state space geometry of chaotic kuramoto-sivashinsky flow · kuramoto-sivashinsky, exhibits, when...

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ON THE STATE SPACE GEOMETRY OF THEKURAMOTO-SIVASHINSKY FLOW IN A PERIODIC DOMAIN

PREDRAG CVITANOVIC∗, RUSLAN L. DAVIDCHACK†, AND EVANGELOS SIMINOS∗

Abstract. The continuous and discrete symmetries of the Kuramoto-Sivashinsky system re-stricted to a spatially periodic domain play a prominent role in shaping the invariant sets of itschaotic dynamics. The continuous spatial translation symmetry leads to relative equilibrium (travel-ing wave) and relative periodic orbit (modulated traveling wave) solutions. The discrete symmetrieslead to existence of equilibrium and periodic orbit solutions, induce decomposition of state spaceinto invariant subspaces, and enforce certain structurally stable heteroclinic connections betweenequilibria. We show, on the example of a particular small-cell Kuramoto-Sivashinsky system, howthe geometry of its dynamical state space is organized by a rigid ‘cage’ built by heteroclinic connec-tions between equilibria, and demonstrate the preponderance of unstable relative periodic orbits andtheir likely role as the skeleton underpinning spatiotemporal turbulence in systems with continuoussymmetries. We also offer novel visualizations of the high-dimensional Kuramoto-Sivashinsky statespace flow through projections onto low-dimensional, PDE representation independent, dynamicallyinvariant intrinsic coordinate frames, as well as in terms of the physical, symmetry invariant energytransfer rates.

Key words. relative periodic orbits, chaos, turbulence, continuous symmetry, Kuramoto-Sivashinsky equation

AMS subject classifications. 35B05, 35B10, 37L05, 37L20, 76F20, 65H10, 90C53

1. Introduction. Recent experimental and theoretical advances [24] support adynamical vision of turbulence: For any finite spatial resolution, a turbulent flowfollows approximately for a finite time a pattern belonging to a finite alphabet ofadmissible patterns. The long term dynamics is a walk through the space of theseunstable patterns. The question is how to characterize and classify such patterns?Here we follow the seminal Hopf paper [26], and visualize turbulence not as a sequenceof spatial snapshots in turbulent evolution, but as a trajectory in an infinite-dimens-ional state space in which an instant in turbulent evolution is a unique point. Inthe dynamical systems approach, theory of turbulence for a given system, with givenboundary conditions, is given by (a) the geometry of the state space and (b) theassociated natural measure, that is, the likelihood that asymptotic dynamics visits agiven state space region.

We pursue this program in context of the Kuramoto-Sivashinsky (KS) equa-tion, one of the simplest physically interesting spatially extended nonlinear systems.Holmes, Lumley and Berkooz [25] offer a delightful discussion of why this system de-serves study as a staging ground for studying turbulence in full-fledged Navier-Stokesboundary shear flows.

Flows described by partial differential equations (PDEs) are said to be infinite-dimensional because if one writes them down as a set of ordinary differential equations(ODEs), a set of infinitely many ODEs is needed to represent the dynamics of onePDE. Even though their state space is thus infinite-dimensional, the long-time dy-namics of viscous flows, such as Navier-Stokes, and PDEs modeling them, such asKuramoto-Sivashinsky, exhibits, when dissipation is high and the system spatial ex-tent small, apparent ‘low-dimensional’ dynamical behaviors. For some of these theasymptotic dynamics is known to be confined to a finite-dimensional inertial mani-

∗School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA†Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

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http://arxiv.org/abs/0709.2944v3

2 P. CVITANOVIC, R. L. DAVIDCHACK, AND E. SIMINOS

fold, though the rigorous upper bounds on this dimension are not of much use in thepractice.

For large spatial extent the complexity of the spatial motions also needs to betaken into account. The systems whose spatial correlations decay sufficiently fast, andthe attractor dimension and number of positive Lyapunov exponents diverges withsystem size are said [27, 41, 9] to be extensive, ‘spatio-temporally chaotic’ or ‘weaklyturbulent.’ Conversely, for small system sizes the accurate description might requirea large set [19] of coupled ODEs, but dynamics can still be ‘low-dimensional’ in thesense that it is characterized with one or a few positive Lyapunov exponents. Thereis no wide range of scales involved, nor decay of spatial correlations, and the systemis in this sense only ‘chaotic.’

For a subset of physicists and mathematicians who study idealized ‘fully devel-oped,’ ‘homogenous’ turbulence the generally accepted usage is that the ‘turbulent’fluid is characterized by a range of scales and an energy cascade describable by statisti-cal assumptions [15]. What experimentalists, engineers, geophysicists, astrophysicistsactually observe looks nothing like a ‘fully developed turbulence.’ In the physicallydriven wall-bounded shear flows, the turbulence is dominated by unstable coherent

structures, that is, localized recurrent vortices, rolls, streaks and like. The statisticalassumptions fail, and a dynamical systems description from first principles is calledfor [25].

The set of invariant solutions investigated here is embedded into a finite-dimens-ional inertial manifold [13] in a non-trivial, nonlinear way. ‘Geometry’ in the titleof this paper refers to our attempt to systematically triangulate this set in terms ofdynamically invariant solutions (equilibria, periodic orbits, . . .) and their unstablemanifolds, in a PDE representation and numerical simulation algorithm independentway. The goal is to describe a given ‘turbulent’ flow quantitatively, not model itqualitatively by a low-dimensional model. For the case investigated here, the statespace representation dimension d ∼ 102 is set by requiring that the exact invariantsolutions that we compute are accurate to ∼ 10−5.

Here comes our quandary. If we ban the words ‘turbulence’ and ‘spatiotemporalchaos’ from our study of small extent systems, the relevance of what we do to largersystems is obscured. The exact unstable coherent structures we determine pertain notonly to the spatially small ‘chaotic’ systems, but also the spatially large ‘spatiotempo-rally chaotic’ and the spatially very large ‘turbulent’ systems. So, for the lack of moreprecise nomenclature, we take the liberty of using the terms ‘chaos,’ ‘spatiotemporalchaos,’ and ‘turbulence’ interchangeably.

In previous work, the state space geometry and the natural measure for thissystem have been studied [6, 37, 38] in terms of unstable periodic solutions restrictedto the antisymmetric subspace of the KS dynamics.

The focus in this paper is on the role continuous symmetries play in spatiotem-poral dynamics. The notion of exact periodicity in time is replaced by the notionof relative spatiotemporal periodicity, and relative equilibria and relative periodic or-bits here play the role the equilibria and periodic orbits played in the earlier studies.Our search for relative periodic orbits in KS system was inspired by Vanessa Lopezet al. [40] investigation of relative periodic orbits of the Complex Ginzburg-Landauequation. However, there is a vast literature on relative periodic orbits since theirfirst appearance, in Poincare study of the 3-body problem [5, 47], where the Lagrangepoints are the relative equilibria. They arise in dynamics of systems with contin-uous symmetries, such as motions of rigid bodies, gravitational N -body problems,

GEOMETRY OF THE KURAMOTO-SIVASHINSKY FLOW 3

molecules and nonlinear waves. Recently Viswanath [48] has found both relativeequilibria and relative periodic orbits in the plane Couette problem. A Hopf bifurca-tion of a traveling wave [1, 2, 34] induces a small time-dependent modulation. Brownand Kevrekidis [4] study bifurcation branches of periodic orbits and relative periodicorbits in KS system in great detail. For our system size (α = 49.04 in their notation)they identify a periodic orbit branch. In this context relative periodic orbits are re-ferred to as ‘modulated traveling waves.’ For fully chaotic flows we find this notiontoo narrow. We compute 60,000 periodic orbits and relative periodic orbits that arein no sense small ‘modulations’ of other solutions, hence our preference for the wellestablished notion of a ‘relative periodic orbit.’

Building upon the pioneering work of refs. [32, 22, 4], we undertake here a studyof the Kuramoto-Sivashinsky dynamics for a specific system size L = 22, sufficientlylarge to exhibit many of the features typical of ‘turbulent’ dynamics observed inlarge KS systems, but small enough to lend itself to a detailed exploration of theequilibria and relative equilibria, their stable/unstable manifolds, determination of alarge number of relative periodic orbits, and a preliminary exploration of the relationbetween the observed spatiotemporal ‘turbulent’ patterns and the relative periodicorbits.

In presence of a continuous symmetry any solution belongs to a group orbit ofequivalent solutions. The problem: If one is to generalize the periodic orbit theoryto this setting, one needs to understand what is meant by solutions being nearby(shadowing) when each solution belongs to a manifold of equivalent solutions. In aforthcoming publication [45] we resolve this puzzle by implementing symmetry reduc-tion. Here we demonstrate that, for relative periodic orbits visiting the neighborhoodof equilibria, if one picks any particular solution, the universe of all other solutionsis rigidly fixed through a web of heteroclinic connections between them. This insightgarnered from study of a 1-dimensional Kuramoto-Sivashinsky PDE is more remark-able still when applied to the plane Couette flow [19], with 3-d velocity fields and twotranslational symmetries.

The main results presented here are: (a) Dynamics visualized through physical,symmetry invariant observables, such as ‘energy,’ dissipation rate, etc., and throughprojections onto dynamically invariant, PDE-discretization independent state spacecoordinate frames, sect. 3. (b) Existence of a rigid ‘cage’ built by heteroclinic con-nections between equilibria, sect. 4. (c) Preponderance of unstable relative periodicorbits and their likely role as the skeleton underpinning spatiotemporal turbulence insystems with continuous symmetries, sect. 6.

2. Kuramoto-Sivashinsky equation. The Kuramoto-Sivashinsky [henceforthKS] system [36, 46], which arises in the description of stability of flame fronts, reaction-diffusion systems and many other physical settings [32], is one of the simplest nonlinearPDEs that exhibit spatiotemporally chaotic behavior. In the formulation adoptedhere, the time evolution of the ‘flame front velocity’ u = u(x, t) on a periodic domainu(x, t) = u(x + L, t) is given by

ut = F (u) = − 12 (u2)x − uxx − uxxxx , x ∈ [−L/2, L/2] . (2.1)

Here t ≥ 0 is the time, and x is the spatial coordinate. The subscripts x and t denotepartial derivatives with respect to x and t. In what follows we shall state results ofall calculations either in units of the ‘dimensionless system size’ L, or the system sizeL = 2πL. Figure 2.1 presents a typical ‘turbulent’ evolution for KS. All numericalresults presented in this paper are for the system size L = 22/2π = 3.5014 . . ., for


which a structurally stable chaotic attractor is observed (see Figure 4.1). Spatialperiodicity u(x, t) = u(x + L, t) makes it convenient to work in the Fourier space,

u(x, t) =

+∞∑

k=−∞

ak(t)eikx/L , (2.2)

with the 1-dimensional PDE (2.1) replaced by an infinite set of ODEs for the complexFourier coefficients ak(t):

ak = vk(a) = (q2k − q4

k) ak − iqk

2

+∞∑

m=−∞

amak−m , (2.3)

where qk = k/L. Since u(x, t) is real, ak = a∗−k, and we can replace the sum by an

m > 0 sum.Due to the hyperviscous damping uxxxx, long time solutions of KS equation are

smooth, ak drop off fast with k, and truncations of (2.3) to 16 ≤ N ≤ 128 terms yieldaccurate solutions for system sizes considered here (see appendix A). Robustness ofthe long-time dynamics of KS as a function of the number of Fourier modes kept intruncations of (2.3) is, however, a subtle issue. Adding an extra mode to a truncationof the system introduces a small perturbation in the space of dynamical systems.However, due to the lack of structural stability both as a function of truncationN , and the system size L, a small variation in a system parameter can (and oftenwill) throw the dynamics into a different asymptotic state. For example, asymptoticattractor which appears to be chaotic in a N -dimensional state space truncation cancollapse into an attractive cycle for (N +1)-dimensions. Therefore, the selection ofparameter L for which a structurally stable chaotic dynamics exists and can be studiedis rather subtle. We have found that the value of L = 22 studied in sect. 4 satisfiesthese requirements. In particular, all of the equilibria and relative equilibria persistand remain unstable when N is increased from 32 (the value we use in our numericalinvestigations) to 64 and 128. Nearly all of the relative periodic orbits we have foundfor this system also exist and remain unstable for larger values of N as well as smallervalues of the integration step size (see appendix C for details).

2.1. Symmetries of Kuramoto-Sivashinsky equation. The KS equation isGalilean invariant: if u(x, t) is a solution, then u(x − ct, t) − c, with c an arbitraryconstant speed, is also a solution. Without loss of generality, in our calculations weshall set the mean velocity of the front to zero,

∫

dxu = 0 . (2.4)

As a0 = 0 in (2.3), a0 is a conserved quantity fixed to a0 = 0 by the condition(2.4). G, the group of actions g ∈ G on a state space (reflections, translations,etc.) is a symmetry of the KS flow (2.1) if g ut = F (g u). The KS equation istime translationally invariant, and space translationally invariant on a periodic do-main under the 1-parameter group of O(2) : τℓ/L, R. If u(x, t) is a solution, thenτℓ/L u(x, t) = u(x + ℓ, t) is an equivalent solution for any shift −L/2 < ℓ ≤ L/2, as isthe reflection (‘parity’ or ‘inversion’)

R u(x) = −u(−x) . (2.5)


Fig. 2.1. A typical spatiotemporally chaotic solution of the Kuramoto-Sivashinsky equation,system size L = 20π

√2 ≈ 88.86. The x coordinate is scaled with the most unstable wavelength 2π

√2,

which is approximately also the mean wavelength of the turbulent flow. The color bar indicates thecolor scheme for u(x, t), used also for the subsequent figures of this type.

The translation operator action on the Fourier coefficients (2.2), represented here bya complex valued vector a = ak ∈ C | k = 1, 2, . . ., is given by

τℓ/L a = g(ℓ) a , (2.6)

where g(ℓ) = diag(eiqk ℓ) is a complex valued diagonal matrix, which amounts tothe k-th mode complex plane rotation by an angle k ℓ/L. The reflection acts on theFourier coefficients by complex conjugation,

R a = −a∗ . (2.7)

Reflection generates the dihedral subgroup D1 = 1, R of O(2). Let U be thespace of real-valued velocity fields periodic and square integrable on the intervalΩ = [−L/2, L/2],

U = u ∈ L2(Ω) | u(x) = u(x + L) . (2.8)

A continuous symmetry maps each state u ∈ U to a manifold of functions withidentical dynamic behavior. Relation R2 = 1 induces linear decomposition u(x) =u+(x) + u−(x), u±(x) = P±u(x) ∈ U±, into irreducible subspaces U = U+ ⊕ U−,where

P+ = (1 + R)/2 , P− = (1 − R)/2 , (2.9)

are the antisymmetric/symmetric projection operators. Applying P+, P− on the KSequation (2.1) we have [32]

u+t = −(u+u+

x + u−u−x ) − u+

xx − u+xxxx

u−t = −(u+u−

x + u−u+x ) − u−

xx − u−xxxx . (2.10)


If u− = 0, KS flow is confined to the antisymmetric U+ subspace,

u+t = −u+u+

x − u+xx − u+

xxxx , (2.11)

but otherwise the nonlinear terms in (2.10) mix the two subspaces.Any rational shift τ1/mu(x) = u(x + L/m) generates a discrete cyclic subgroup

Cm of O(2), also a symmetry of KS system. Reflection together with Cm generatesanother symmetry of KS system, the dihedral subgroup Dm of O(2). The only non-zero Fourier components of a solution invariant under Cm are ajm 6= 0, j = 1, 2, · · · ,while for a solution invariant under Dm we also have the condition Re aj = 0 for allj. Dm reduces the dimensionality of state space and aids computation of equilibriaand periodic orbits within it. For example, the 1/2-cell translations

τ1/2 u(x) = u(x + L/2) (2.12)

and reflections generate O(2) subgroup D2 = 1, R, τ, τR, which reduces the statespace into four irreducible subspaces (for brevity, here τ = τ1/2):

τ R τR

P (1) =1

4(1 + τ + R + τR) S S S

P (2) =1

4(1 + τ − R − τR) S A A

P (3) =1

4(1 − τ + R − τR) A S A (2.13)

P (4) =1

4(1 − τ − R + τR) A A S .

P (j) is the projection operator onto u(j) irreducible subspace, and the last 3 columnsrefer to the symmetry (or antisymmetry) of u(j) functions under reflection and 1/2-cell shift. By the same argument that identified (2.11) as the invariant subspace ofKS, here the KS flow stays within the US = U(1) + U(2) irreducible D1 subspace of uprofiles symmetric under 1/2-cell shifts.

While in general the bilinear term (u2)x mixes the irreducible subspaces of Dn,for D2 there are four subspaces invariant under the flow [32]:0: the u(x) = 0 equilibriumU+ = U(1) + U(3): the reflection D1 irreducible space of antisymmetric u(x)U

S = U(1) + U

(2): the shift D1 irreducible space of L/2 shift symmetric u(x)U(1): the D2 irreducible space of u(x) invariant under x 7→ L/2 − x, u 7→ −u.

With the continuous translational symmetry eliminated within each subspace, thereare no relative equilibria and relative periodic orbits, and one can focus on the equi-libria and periodic orbits only, as was done for U+ in refs. [6, 37, 38]. In the Fourierrepresentation, the u ∈ U+ antisymmetry amounts to having purely imaginary coeffi-cients, since a−k = a∗

k = −ak. The 1/2 cell-size shift τ1/2 generated 2-element discretesubgroup 1, τ1/2 is of particular interest because in the U+ subspace the transla-tional invariance of the full system reduces to invariance under discrete translation(2.12) by half a spatial period L/2.

Each of the above dynamically invariant subspaces is unstable under small per-turbations, and generic solutions of Kuramoto-Sivashinsky equation belong to the fullspace. Nevertheless, since all equilibria of the KS flow studied in this paper lie inthe U+ subspace (see sect. 4), U+ plays important role for the global geometry of theflow. The linear stability matrices of these equilibria have eigenvectors both in andoutside of U+, and need to be computed in the full state space.


2.2. Equilibria and relative equilibria. Equilibria (or the steady solutions)are the fixed profile time-invariant solutions,

u(x, t) = uq(x) . (2.14)

Due to the translational symmetry, the KS system also allows for relative equilibria(traveling waves, rotating waves), characterized by a fixed profile uq(x) moving withconstant speed c, that is

u(x, t) = uq(x − ct) . (2.15)

Here suffix q labels a particular invariant solution. Because of the reflection symmetry(2.5), the relative equilibria come in counter-traveling pairs uq(x− ct), −uq(−x+ ct).

The relative equilibrium condition for the Kuramoto-Sivashinsky PDE (2.1) is theODE

12 (u2)x + uxx + uxxxx = c ux (2.16)

which can be analyzed as a dynamical system in its own right. Integrating once weget

12u2 − cu + ux + uxxx = E . (2.17)

This equation can be interpreted as a 3-dimensional dynamical system with spatial co-ordinate x playing the role of ‘time,’ and the integration constant E can be interpretedas ‘energy,’ see sect. 3.

For E > 0 there is rich E-dependent dynamics, with fractal sets of boundedsolutions investigated in depth by Michelson [42]. For L < 1 the only equilibrium ofthe system is the globally attracting constant solution u(x, t) = 0, denoted E0 fromnow on. With increasing system size L the system undergoes a series of bifurcations.The resulting equilibria and relative equilibria are described in the classical papersof Kevrekidis, Nicolaenko and Scovel [32], and Greene and Kim [22], among others.The relevant bifurcations up to the system size investigated here are summarized inFigure 2.2: at L = 22/2π = 3.5014 · · · , the equilibria are the constant solution E0, theequilibrium E1 called GLMRT by Greene and Kim [39, 22], the 2- and 3-cell statesE2 and E3, and the pairs of relative equilibria TW±1, TW±2. All equilibria are in theantisymmetric subspace U

+, while E2 is also invariant under D2 and E3 under D3.In the Fourier representation the relative equilibria time dependence is

ak(t)e−itcqk = ak(0) . (2.18)

Differentiating with respect to time, we obtain the Fourier space version of the relativeequilibrium condition (2.16),

vk(a) − iqkcak = 0 , (2.19)

which we solve for (time independent) ak and c. Periods of spatially periodic equilib-ria are L/n with integer n. Every time the system size crosses L = n, n-cell states aregenerated through pitchfork bifurcations off u = 0 equilibrium. Due to the transla-tional invariance of Kuramoto-Sivashinsky equation, they form invariant circles in thefull state space. In the U+ subspace considered here, they correspond to 2n points,each shifted by L/2n. For a sufficiently small L the number of equilibria is small andconcentrated on the low wave-number end of the Fourier spectrum.


0.5 1. 1.5 2. 2.5 3. 3.5 4.

0.2

0.4

0.6

0.8

1.

1.2

1.4

1.6

E

L

E0

E1

E2

E3

TW±1

TW±2

Fig. 2.2. The energy (3.6) of the equilibria and relative equilibria that exist up to L = 22,L = 3.5014 . . ., plotted as a function of the system size L = L/2π (additional equilibria, not presentat L = 22 are given in ref. [22]). Solid curves denote n-cell solutions E2 and E3, dotted curves theGLMRT equilibrium E1, and dashed curves the relative equilibria TW±1 and TW±2. The parameter

α of refs. [32, 22] is related to the system size by L =p

α/4.

In a periodic box of size L both equilibria and relative equilibria are periodicsolutions embedded in 3-d space, conveniently represented as loops in (u, ux, uxx)space, see Figure 5.1 (d). In this representation the continuous translation symmetryis automatic – a rotation in the [0, L] periodic domain only moves the points alongthe loop. For an equilibrium the points are stationary in time; for relative equilibriumthey move in time, but in either case, the loop remains invariant. So we do not havethe problem that we encounter in the Fourier representation, where seen from theframe of one of the equilibria the rest trace out circles under the action of continuoussymmetry translations.

From (2.3) we see that the origin u(x, t) = 0 has Fourier modes as the linearstability eigenvectors (see appendix B). The |k| < L long wavelength perturbations ofthe flat-front equilibrium are linearly unstable, while for |k| sufficiently larger than Lthe short wavelength perturbations are strongly contractive. The high k eigenvalues,corresponding to rapid variations of the flame front, decay so fast that the correspond-ing eigendirections are physically irrelevant. Indeed, ref. [49] shows that the chaoticsolutions of spatially extended dissipative systems evolve within an inertial manifoldspanned by a finite number of physical modes, hyperbolically isolated from a set ofresidual degrees of freedom with high k, themselves individually isolated from eachother. The most unstable mode, nearest to |k| = L/

√2, sets the scale of the mean

wavelength√

2 of the KS ‘turbulent’ dynamics, see Figure 2.1.

2.3. Relative periodic orbits, symmetries and periodic orbits. The KSequation (2.1) is time translationally invariant, and space translationally invariantunder the 1-d Lie group of O(2) rotations: if u(x, t) is a solution, then u(x + ℓ, t) and−u(−x, t) are equivalent solutions for any −L/2 < ℓ ≤ L/2. As a result of invarianceunder τℓ/L, KS equation can have relative periodic orbit solutions with a profile up(x),period Tp, and a nonzero shift ℓp

τℓp/Lu(x, Tp) = u(x + ℓp, Tp) = u(x, 0) = up(x) . (2.20)


Relative periodic orbits (2.20) are periodic in cp = ℓp/Tp co-rotating frame (see Fig-ure 8.3), but in the stationary frame their trajectories are quasiperiodic. Due to thereflection symmetry (2.5) of KS equation, every relative periodic orbit up(x) with shiftℓp has a symmetric partner −up(−x) with shift −ℓp.

Due to invariance under reflections, KS equation can also have relative periodicorbits with reflection, which are characterized by a profile up(x) and period Tp

Ru(x + ℓ, Tp) = −u(−x − ℓ, Tp) = u(x + ℓ, 0) = up(x) , (2.21)

giving the family of equivalent solutions parameterized by ℓ (as the choice of thereflection point is arbitrary, the shift can take any value in −L/2 < ℓ ≤ L/2).

Armbruster et al. [2, 1] and Brown and Kevrekidis [4] (see also ref. [34]) linkthe birth of relative periodic orbits to an infinite period global bifurcation involvinga heteroclinic loop connecting equilibria or a bifurcation of relative equilibria, andalso report creation of relative periodic orbit branches through bifurcation of periodicorbits.

As ℓ is continuous in the interval [−L/2, L/2], the likelihood of a relative periodicorbit with ℓp = 0 shift is zero, unless an exact periodicity is enforced by a discretesymmetry, such as the dihedral symmetries discussed above. If the shift ℓp of a relativeperiodic orbit with period Tp is such that ℓp/L is a rational number, then the orbitis periodic with period nTp. The likelihood to find such periodic orbits is also zero.

However, due to the KS equation invariance under the dihedral Dn and cyclic Cn

subgroups, the following types of periodic orbits are possible:

(a) The periodic orbit lies within a subspace pointwise invariant under the actionof Dn or Cn. For instance, for D1 this is the U+ antisymmetric subspace, −up(−x) =up(x), and u(x, Tp) = u(x, 0) = up(x). The periodic orbits found in refs. [6, 38]are all in U

+, as the dynamics is restricted to antisymmetric subspace. For L = 22the dynamics in U+ is dominated by attracting (within the subspace) heteroclinicconnections and thus we have no periodic orbits of this type, or in any other of theDn–invariant subspaces, see sect. 4.

(b) The periodic orbit satisfies

u(x, t + Tp) = γu(x, t) , (2.22)

for some group element γ ∈ O(2) such that γm = e for some integer m so that theorbit repeats after time mTp (see ref. [21] for a general discussion of conditions on thesymmetry of a periodic orbit). If an orbit is of reflection type (2.21), Rτℓ/Lu(x, Tp) =−u(−x − ℓ, Tp) = u(x, 0), then it is pre-periodic to a periodic orbit with period 2Tp.Indeed, since (Rτℓ/L)2 = R2 = 1, and the KS solutions are time translation invariant,it follows from (2.21) that

u(x, 2Tp) = Rτℓ/Lu(x, Tp) = (Rτℓ/L)2u(x, 0) = u(x, 0) .

Thus any shift acquired during time 0 to Tp is compensated by the opposite shiftduring evolution from Tp to 2Tp. All periodic orbits we have found for L = 22 areof type (2.22) with γ = R. Pre-periodic orbits with γ ∈ Cn have been found byBrown and Kevrekidis [4] for KS system sizes larger than ours, but we have not foundany for L = 22. Pre-periodic orbits are a hallmark of any dynamical system with adiscrete symmetry, where they have a natural interpretation as periodic orbits in thefundamental domain [12, 11].


3. Energy transfer rates. In physical settings where the observation times aremuch longer than the dynamical ‘turnover’ and Lyapunov times (statistical mechan-ics, quantum physics, turbulence) periodic orbit theory [11] provides highly accuratepredictions of measurable long-time averages such as the dissipation and the turbu-lent drag [19]. Physical predictions have to be independent of a particular choice ofODE representation of the PDE under consideration and, most importantly, invariantunder all symmetries of the dynamics. In this section we discuss a set of such physicalobservables for the 1-d KS invariant under reflections and translations. They offer arepresentation of dynamics in which the symmetries are explicitly quotiented out. Weshall use these observables in sect. 8 in order to visualize a set of solutions on thesecoordinates.

The space average of a function a = a(x, t) = a(u(x, t)) on the interval L,

〈a〉 =1

L

∮

dx a(x, t) , (3.1)

is in general time dependent. Its mean value is given by the time average

a = limt→∞

1

t

∫ t

0

dτ 〈a〉 = limt→∞

1

t

∫ t

0

1

L

∮

dτ dx a(x, τ) . (3.2)

The mean value of a = a(uq) ≡ aq evaluated on equilibrium or relative equilibriumu(x, t) = uq(x − ct), labeled by q as in (2.15), is

aq = 〈a〉q = aq . (3.3)

Evaluation of the infinite time average (3.2) on a function of a periodic orbit or relativeperiodic orbit up(x, t) = up(x + ℓp, t + Tp) requires only a single Tp traversal,

ap =1

Tp

∫ Tp

0

dτ 〈a〉 . (3.4)

Equation (2.1) can be written as

ut = −Vx , V (x, t) = 12u2 + ux + uxxx . (3.5)

If u is ‘flame-front velocity’ then E, defined in (2.17), can be interpreted as the meanenergy density. So, even though KS is a phenomenological small-amplitude equation,the time-dependent L2 norm of u,

E =1

L

∮

dxV (x, t) =1

L

∮

dxu2

2, (3.6)

has a physical interpretation [22] as the average ‘energy’ density of the flame front.This analogy to the mean kinetic energy density for the Navier-Stokes motivates whatfollows.

The energy (3.6) is intrinsic to the flow, independent of the particular ODE basisset chosen to represent the PDE. However, as the Fourier amplitudes are eigenvectorsof the translation operator, in the Fourier space the energy is a diagonalized quadraticnorm,

E =∞∑

k=−∞

Ek , Ek = 12 |ak|2 , (3.7)


and explicitly invariant term by term under translations (2.6) and reflections (2.5).Take time derivative of the energy density (3.6), substitute (2.1) and integrate by

parts. Total derivatives vanish by the spatial periodicity on the L domain:

E = 〈ut u〉 = −⟨(

u2/2 + ux + uxxx

)

xu⟩

=⟨

ux u2/2 + u2x + ux uxxx

⟩

. (3.8)

The first term in (3.8) vanishes by integration by parts, 3⟨

ux u2⟩

=⟨

(u3)x

⟩

= 0 , andintegrating the third term by parts yet again one gets [22] that the energy variation

E = P − D , P =⟨

u2x

⟩

, D =⟨

u2xx

⟩

(3.9)

balances the power P pumped in by anti-diffusion uxx against the energy dissipationrate D by hyper-viscosity uxxxx in the KS equation (2.1).

The time averaged energy density E computed on a typical orbit goes to a con-stant, so the mean values (3.2) of drive and dissipation exactly balance each other:

E = limt→∞

1

t

∫ t

0

dτ E = P − D = 0 . (3.10)

In particular, the equilibria and relative equilibria fall onto the diagonal in Fig-ure 8.1 (a), and so do time averages computed on periodic orbits and relative periodicorbits:

Ep =1

Tp

∫ Tp

0

dτ E(τ) , P p =1

Tp

∫ Tp

0

dτ P (τ) = Dp . (3.11)

In the Fourier basis (3.7) the conservation of energy on average takes form

0 =

∞∑

k=−∞

(q2k − q4

k)Ek , Ek(t) = 12 |ak(t)|2 . (3.12)

The large k convergence of this series is insensitive to the system size L; Ek haveto decrease much faster than q−4

k . Deviation of Ek from this bound for small kdetermines the active modes. For equilibria an L-independent bound on E is givenby Michelson [42]. The best current bound [17, 3] on the long-time limit of E as afunction of the system size L scales as E ∝ L2.

4. Geometry of state space with L = 22. We now turn to exploring Hopf’svision numerically, on a specific KS system. An instructive example is offered by thedynamics for the L = 22 system that we specialize to for the rest of this paper. Thesize of this small system is ∼ 2.5 mean wavelengths (L/

√2 = 2.4758 . . .), and the

competition between states with wavenumbers 2 and 3 leads to what, in the contextof boundary shear flows, would be called [23] the ‘empirically observed sustained tur-bulence,’ but in the present context may equally well be characterized as a ‘chaoticattractor.’ A typical long orbit is shown in Figure 4.1. Asymptotic attractor struc-ture of small systems like the one studied here is very sensitive to system parametervariations, and, as is true of any realistic unsteady flow, there is no rigorous way ofestablishing that this ‘turbulence’ is sustained for all time, rather than being merelya very long transient on a way to an attracting periodic state. For large system size,as the one shown in Figure 2.1, it is hard to imagine a scenario under which at-tracting periodic states (as shown in ref. [16], they do exist) would have significantly


Fig. 4.1. A typical chaotic orbit of the KS flow, system size L = 22.

large immediate basins of attraction. Regardless of the (non)existence of a t → ∞chaotic attractor, study of the invariant unstable solutions and the associated Smalehorseshoe structures in system’s state space offers valuable insights into the observedunstable ‘coherent structures.’

Because of the strong k4 contraction, for a small system size the long-time dy-namics is confined to low-dimensional inertial manifold [29]. Indeed, numericallythe covariant Lyapunov vectors [20] of the L = 22 chaotic attractor separate into8 “physical” vectors with small Lyapunov exponents (λj) = (0.048, 0, 0, −0.003,−0.189, −0.256, −0.290, −0.310), and the remaining 54 “hyperbolically isolated”vectors with rapidly decreasing exponents (λj) = (−1.963, −1.967, −5.605, −5.605,

−11.923, −11.923, · · · ) ≈ −(j/L)4, in full agreement with the Yang et al. [49] investi-gations of KS for large systems sizes. The chaotic dynamics mostly takes place close toa 8-dimensional manifold, with strong contraction in other dimensions. The two zeroexponents are due to the time and space translational symmetries of the Kuramoto-Sivashinsky equation and the 2 corresponding dimensions can be quotiented out bymeans of discrete-time Poincare sections and O(2) group orbit slices. It was shown inrefs. [6, 38] that within unstable-manifold curvilinear coordinate frames, the dynam-ics on the attractor can sometimes be well approximated by local 1- or 2-dimensionalPoincare return maps. Hence a relatively small number of real Fourier modes, suchas 62 to 126 used in calculations presented here, suffices to obtain invariant solutionsnumerically accurate to within 10−5.

We next investigate the properties of equilibria and relative equilibria and deter-mine numerically a large set of the short periods relative periodic orbits for KS in aperiodic cell of size L = 22.

5. Equilibria and relative equilibria for L = 22. In addition to the trivialequilibrium u = 0 (denoted E0), we find three equilibria with dominant wavenumberk (denoted Ek) for k = 1, 2, 3. All equilibria, shown in Figure 5.1, are symmetric withrespect to the reflection symmetry (2.5). In addition, E2 and E3 are symmetric withrespect to translation (2.12), by L/2 and L/3, respectively. E2 and E3 essentially liein the 2nd and 3rd Fourier component complex planes, with small deformations of the


(a)10 0 10

x

1.5

0.

1.5

u

(b)10 0 10

x

1.5

0.

1.5

u

(c)10 0 10

x

1.5

0.

1.5

u

(d)

20

2u

2

0

2ux

2

0

2

uxx

20

2u

2

0

2ux

Fig. 5.1. (a) E1, (b) E2, and (c) E3 equilibria. The E0 equilibrium is the u(x) = 0 solution.(d) (u, ux, uxx) representation of (red) E1, (green) E2, (blue) E3 equilibria, (purple) TW+1, and(orange) TW−1 relative equilibria. L = 22 system size.

k = 2j and k = 3j harmonics, respectively.

The stability of the equilibria is characterized by the eigenvalues λj of the stabilitymatrix. The leading 10 eigenvalues for each equilibrium are listed in Table 5.1; thosewith µ > −2.5 are also plotted in Figure 5.2. We have computed (available uponrequest) the corresponding eigenvectors as well. As an equilibrium with Re λj > 0is unstable in the direction of the corresponding eigenvector e(j), the eigenvectorsprovide flow-intrinsic (PDE discretization independent) coordinates which we use forvisualization of unstable manifolds and homo/heteroclinic connections between equi-libria. We find such coordinate frames, introduced by Gibson et al. [19, 18], bettersuited to visualization of nontrivial solutions than the more standard Fourier mode(eigenvectors of the u(x, t) = 0 solution) projections.

The eigenvalues of E0 are determined by the linear part of the KS equation (B.4):λk = (k/L)2 − (k/L)4. For L = 22, there are three pairs of unstable eigenvalues,corresponding, in decreasing order, to three unstable modes k = 2, 3, and 1. For eachmode, the corresponding eigenvectors lie in the plane spanned by Re ak and Im ak.Table 5.1 lists the symmetries of the stability eigenvectors of equilibria E1 to E3.

Consistent with the bifurcation diagram of Figure 2.2, we find two pairs of relativeequilibria (2.15) with velocities c = ±0.73699 and ±0.34954 which we label TW±1

and TW±2, for ‘traveling waves.’ The profiles of the two relative equilibria and theirtime evolution with eventual decay into the chaotic attractor are shown in Figure 5.3.The leading eigenvalues of TW±1 and TW±2 are listed in Table 5.1.

Table 5.2 lists equilibrium energy E, the local Poincare section return time T ,radially expanding Floquet multiplier Λe, and the least contracting Floquet multiplierΛc for all L = 22 equilibria and relative equilibria. The return time T = 2π/νe is given


2 1.5 1 0.5 0i

0.3

0.2

0.1

0

0.1

0.2

0.3

i

E3

E2

E1

E0

Fig. 5.2. Leading equilibrium stability eigenvalues, L = 22 system size.

by the imaginary part of the leading complex eigenvalue, the expansion multiplier perone turn of the most unstable spiral-out by Λe ≈ exp(µeT ), and the contraction ratealong the slowest contracting stable eigendirection by Λc ≈ exp(µcT ). For E3 andTW±2, whose leading eigenvalues are real, we use T = 1/λ1 as the characteristic timescale. While the complex eigenvalues set time scales of recurrences, this time scale isuseful for comparison of leading expanding and the slowest contracting multiplier. Welearn that the shortest ‘turn-over’ time is ≈ 10− 20, and that if there exist horseshoesets of unstable periodic orbits associated with these equilibria, they have unstablemultipliers of order of Λe ∼ 5 − 10, and that they are surprisingly thin in the foldingdirection, with contracting multipliers of order of 10−2, as also observed in ref. [38].

5.1. Unstable manifolds of equilibria and their heteroclinic connec-tions. As shown in Table 5.1, the E1 equilibrium has two unstable planes withinwhich the solutions are spiralling out (that is, two pairs of complex conjugate eigen-values). The E2 has one such plane, while the E3 has two real positive eigenvalues, sothe solutions are moving radially away from the equilibrium within the plane spannedby the corresponding eigenvectors. Since E1 has a larger unstable subspace, it is ex-pected to have much less influence on the long time dynamics compared to E2 andE3.

Many methods have been developed for visualization of stable and unstable mani-folds, see ref. [33] for a survey. For high-dimensional contracting flows visualization ofstable manifolds is impossible, unless the system can be restricted to an approximatelow-dimensional inertial manifold, as, for example, in ref. [28]. The unstable manifoldvisualization also becomes harder as its dimension increases. Here we concentrate onvisualizations of 1– and 2–dimensional unstable manifolds. Our visualization is unso-phisticated compared to the methods of ref. [33], yet sufficient for our purposes since,as we shall see, the unstable manifolds we study terminate in another equilibrium andthus there is no need to track them for long times.

To construct an invariant manifold containing solutions corresponding to the pairof unstable complex conjugate eigenvalues, λ = µ ± iν, µ > 0, we start with a set of


Table 5.1

Leading eigenvalues λj = µj ± iνj and symmetries of the corresponding eigenvectors of KSequilibria and relative equilibria for L = 22 system size. We have used as our reference statesthe ones that lie within the antisymmetric subspace U+, and also listed the symmetries of the L/4translated ones.

E1 µj νj Symmetry τ1/4En Symmetry

λ1,2 0.1308 0.3341 - -

λ3,4 0.0824 0.3402 U+ U(1)

λ5 0 - -

λ6,7 −0.2287 0.1963 U+

U(1)

λ8 −0.2455 - -

λ9 −2.0554 U+ U(1)

λ10 −2.0619 - -

E2

λ1,2 0.1390 0.2384 U+

U(1)

λ3 0 τ1/2 τ1/2

λ4,5 −0.0840 0.1602 U(1) U+

λ6 −0.1194 τ1/2 τ1/2

λ7,8 −0.2711 0.3563 U+, U(1), τ1/2 U+, U(1), τ1/2

λ9 −2.0130 U(1)

U+

λ10 −2.0378 U+ U(1)

E3

λ1 0.0933 U+ U(1)

λ2 0.0933 - -λ3 0 τ1/3 τ1/3

λ4 −0.4128 U+, τ1/3 U(1), τ1/3

λ5,6 −0.6108 0.3759 U+ U(1)

λ7,8 −0.6108 0.3759 - -λ9 −1.6641 - -

λ10 −1.6641 U+ U(1)

TW±1

λ1,2 0.1156 0.8173 - -λ3,4 0.0337 0.4189 - -λ5 0 - -λ6 −0.2457 - -

λ7,8 −0.3213 0.9813 - -

TW±2

λ1 0.3370 - -λ2 0 - -

λ3,4 −0.0096 0.6288 - -λ5,6 −0.2619 0.5591 - -λ7,8 −0.3067 0.0725 - -

initial conditions near equilibrium Ek,

a(0) = aEk+ ǫ exp(δ)e(j) , (5.1)

where δ takes a set of values uniformly distributed in the interval [0, 2πµ/ν], e(j) is aunit vector in the unstable plane, and ǫ > 0 is small.

The manifold starting within the first unstable plane of E1, with eigenvalues0.1308 ± i 0.3341, is shown in Figure 5.4. It appears to fall directly into the chaoticattractor. The behavior of the manifold starting within the second unstable plane ofE1, eigenvalues 0.0824± i 0.3402, is remarkably different: as can be seen in Figure 5.5,almost all orbits within the manifold converge to the equilibrium E2. The manifold


Fig. 5.3. Relative equilibria: TW+1 with velocity c = 0.737 and TW+2 with velocity c = 0.350.The upper panels show the relative equilibria profiles. The lower panels show evolution of slightlyperturbed relative equilibria and their decay into generic turbulence. Each relative equilibrium has areflection symmetric partner related by u(x) → −u(−x) travelling with velocity −c.

Table 5.2

Properties of equilibria and relative equilibria determining the system dynamics in their vicin-ity. T is characteristic time scale of the dynamics, Λe and Λc are the leading expansion andcontraction multipliers, and E is the energy (3.6).

E T Λe Λc

E1 0.2609 18.81 11.70 0.01E2 0.4382 26.35 39.00 0.11E3 1.5876 10.72 2.72 0.01TW±1 0.4649 7.69 2.43 0.15TW±2 0.6048 2.97 2.72 0.97

also contains a heteroclinic connection from E1 to E3, and is bordered by the λ1-eigendirection unstable manifold of E3.

The two-dimensional unstable manifold of E2 is shown in Figure 5.6. All orbitswithin the manifold, except for the heteroclinic connections from E2 to E3, convergeto E2 shifted by L/4, so this manifold, minus the heteroclinic connections, can beviewed as a homoclinic connection.

The equilibrium E3 has a pair of real unstable eigenvalues equal to each other.Therefore, within the plane spanned by the corresponding eigenvectors, the orbitsmove radially away from the equilibrium. In order to trace out the unstable manifold,


Fig. 5.4. The left panel shows the unstable manifold of equilibrium E1 starting within theplane corresponding to the first pair of unstable eigenvalues. The coordinate axes v1, v2, and v3 areprojections onto three orthonormal vectors v1, v2, and v3, respectively, constructed from vectorsRe e(1), Im e(1), and Re e(6) by Gram-Schmidt orthogonalization. The right panel shows spatialrepresentation of two orbits A and B. The change of color from blue to red indicates increasingvalues of u(x), as in the colorbar of Figure 2.1.

Fig. 5.5. The left panel shows the unstable manifold of equilibrium E1 starting within the planecorresponding to the second pair of unstable eigenvalues. The coordinate axes v1, v2, and v3 areprojections onto three orthonormal vectors v1, v2, and v3, respectively, constructed from vectorsRe e(3), Im e(3), and Re e(6) by Gram-Schmidt orthogonalization. The right panel shows spatialrepresentation of three orbits. Orbits B and C pass close to the equilibrium E3.

we start with a set of initial conditions within the unstable plane

a(0) = aE3 + ǫ(v1 cosφ + v2 sinφ) , φ ∈ [0, 2π] , (5.2)

where v1 and v2 are orthonormal vectors within the plane spanned by the two un-stable eigenvectors. The unstable manifold of E3 is shown in Figure 5.8. The 3-foldsymmetry of the manifold is related to the symmetry of E3 with respect to translationby L/3. The manifold contains heteroclinic orbits connecting E3 to three differentpoints of the circle of equilibria E2 translated set of solutions. Note also that thesegments of orbits B and C between E3 and E2 in Figures 5.5 and 5.6 represent thesame heteroclinic connections as orbits B and C in Figure 5.8.

Heteroclinic connections are non-generic for high-dimensional systems, but can be


Fig. 5.6. The left panel shows the two-dimensional unstable manifold of equilibrium E2. Thecoordinate axes v1, v2, and v3 are projections onto three orthonormal vectors v1, v2, and v3,respectively, constructed from vectors Re e(1), Im e(1), and Re e(7) by Gram-Schmidt orthogo-nalization. The right panel shows spatial representation of three orbits. Orbits B and C pass closeto the equilibrium E3. See Figure 5.7 for a different visualization.

(a) (b)

Fig. 5.7. (a) (blue/green) The unstable manifold of E2 equilibrium, projection in the coordinateaxes of Figure 5.6. (black line) The circle of E2 equilibria related by the translation invariance.(purple line) The circle of E3 equilibria. (red) The heteroclinic connection from the E2 equilibriumto the E3 equilibrium splits the manifold into two parts, colored (blue) and (green). (b) E2 equilibriumto E3 equilibrium heteroclinic connection, (Re e(2),Re e(3), (Im e(2)+Im e(3))/

√2) projection. Here

we omit the unstable manifold of E2, keeping only a few neighboring trajectories in order to indicatethe unstable manifold of E3. The E2 and E3 families of equilibria arising from the continuoustranslational symmetry of KS on a periodic domain are indicated by the two circles.

robust in systems with continuous symmetry, see ref. [35] for a review. Armbruster et

al. [2] study a fourth order truncation of KS dynamics on the center-unstable manifoldof E2 close to a bifurcation off the constant u(x, t) = 0 solution and prove existence ofa heteroclinic connection, see also ref. [1]. Kevrekidis et al. [32] study the dynamicsnumerically and establish the existence of a robust heteroclinic connection for a rangeof parameters close to the onset of the 2-cell branch in terms of the symmetry anda flow invariant subspace. We adopt their arguments to explain the new heteroclinicconnections shown in Figure 5.9 that we have found for L = 22. For our system sizethere are exactly two representatives of the E2 family that lie in the intersection ofU+ and U(1) related to each other by an L/4 shift. Denote them by E2 and τ1/4E2

respectively. The unstable eigenplane of E2 lies on U+ while that of τ1/4E2 lies on

U(1), cf. Table 5.1. The E3 family members that live in U+ have one of their unstableeigenvectors (the one related to the heteroclinic connection to E2 family) on U+,


Fig. 5.8. The left panel shows the two-dimensional unstable manifold of equilibrium E3. Thecoordinate axes v1, v2, and v3 are projections onto three orthonormal vectors v1, v2, and v3,respectively, constructed from vectors e(1), e(2), and e(4) by Gram-Schmidt orthogonalization. Theblack line shows a family of E2 equilibria related by translational symmetry. The right panel showsspatial representation of three orbits. Orbits B and C are two different heteroclinic orbits connectingE3 to the same point on the E2 line.

Fig. 5.9. Heteroclinic connections on U+: (red) The unstable manifold of E1 equilibrium.

(blue/green) The unstable manifold of E2 equilibrium. (black) Heteroclinic connections from E3 equi-librium to τ1/4E2 equilibrium, where τ1/mu(x) = u(x + L/m) is a rational shift (2.6). Projectionfrom 128 dimensions onto the plane given by the vectors aE2

− aτ1/4E2and aE3

− aτ1/2E3.

while the other does not lie on symmetry-invariant subspace. Similarly, for the E1

family we observe that the equilibria in U+ have an unstable plane on U

+ (againrelated to the heteroclinic connection) and a second one with no symmetry. Thusτ1/4E2 appears as a sink on U+, while all other equilibria appear as sources. Thisexplains the heteroclinic connections from E1 , E2 and E3 to τ1/4E2. Observing that

τ1/4U+ = U(1) and taking into account Table 5.1 we understand that within U(1) wehave connections from τ1/4E2 (and members of E1 and E3 families) to E2 and theformation of a heteroclinic loop. Due to the translational invariance of KS there is aheteroclinic loop for any two points of the E2 family related by an τ1/4-shift.


(a) (c) (e) (g) (i) (k)

(b) (d) (f ) (h) (j ) (l)

Fig. 6.1. Selected relative periodic and pre-periodic orbits of KS flow with L = 22: (a) Tp =16.3, ℓp = 2.86; (b) Tp = 32.8, ℓp = 10.96; (c) Tp = 33.5, ℓp = 4.04; (d) Tp = 34.6, ℓp = 9.60;(e) Tp = 47.6, ℓp = 5.68; (f) Tp = 59.9, ℓp = 5.44; (g) Tp = 71.7, ℓp = 5.503; (h) Tp = 84.4,ℓp = 5.513; (i) Tp = 10.3; (j) Tp = 32.4; (k) Tp = 33.4; (l) Tp = 35.2. Horizontal and vertical whitelines indicate periodicity and phase shift of the orbits, respectively.

6. Relative periodic orbits for L = 22. The relative periodic orbits satisfythe condition (2.20) u(x + ℓp, Tp) = u(x, 0), where Tp is the period and ℓp the phaseshift. We have limited our search to orbits with Tp < 200 and found over 30 000relative periodic orbits with ℓp > 0. The details of the algorithm used and the searchstrategy employed are given in appendix C. Each relative periodic orbit with phaseshift ℓp > 0 has a reflection symmetric partner up(x) → −up(−x) with phase shift−ℓp.

The small period relative periodic orbits outline the coarse structure of the chaoticattractor, while the longer period relative periodic orbits resolve the finer details ofthe dynamics. The first four orbits with the shortest periods we have found are shownin Figure 6.1 (a-d). The shortest relative periodic orbit with Tp = 16.4 is also the mostunstable, with one positive Floquet exponent equal 0.328. The other short orbits areless unstable, with the largest Floquet exponent in the range 0.018 – 0.073, typical ofthe long time attractor average.

We have found relative periodic orbits which stay close to the unstable manifoldof E2. As is illustrated in Figure 6.1 (e-h), all such orbits have shift ℓp ≈ L/4,similar to the shift of orbits within the unstable manifold of E2, which start at E2


(a) (b)

Fig. 8.1. (a) Power input P vs. dissipation rate D (b) energy E vs. power input P , forseveral equilibria and relative equilibria, a relative periodic orbit, and a typical ‘turbulent’ long-timetrajectory. Projections of the heteroclinic connections are given in Figure 8.2. System size L = 22.

and converge to τ1/4E2 (see Figure 5.6). This confirms that the ‘cage’ of unstablemanifolds of equilibria plays an important role in organizing the chaotic dynamics ofthe KS equation.

7. Pre-periodic orbits. As discussed in Sect. 2.3, a relative periodic orbit willbe periodic, that is, ℓp = 0, if it either (a) lives within the U

+ antisymmetric subspace,−u(−x, 0) = u(x, 0), or (b) returns to its reflection or its discrete rotation aftera period: u(x, t + Tp) = γu(x, t), γm = e, and is thus periodic with period mTp.The dynamics of KS flow in the antisymmetric subspace and periodic orbits withsymmetry (a) have been investigated previously [6, 37, 38]. The KS flow does nothave any periodic orbits of this type for L = 22.

Using the algorithm and strategy described in appendix C, we have found over30 000 pre-periodic orbits with Tp < 200 which possess the symmetry of type (b) withγ = R ∈ D1. Some of the shortest such orbits we have found are shown in Figure 6.1 (i-l). Several were found as repeats of pre-periodic orbits during searches for relativeperiodic orbits with non-zero shifts, while most have been found as solutions of thepre-periodic orbit condition (2.21) with reflection, which takes form

− g(−ℓ)a∗(Tp) = a(0) . (7.1)

in the Fourier space representation (compare this to the condition (C.1) for relativeperiodic orbits).

8. Energy transfer rates for L = 22. In Figure 8.1 we plot (3.9), the time-dependent E in the power input P vs. dissipation rate D plane, for L = 22 equilibriaand relative equilibria, a selected relative periodic orbit, and for a typical ‘turbulent’long-time trajectory.

Projections from the ∞-dimensional state space onto the 3-dimensional (E, P, D)representation of the flow, such as Figures 8.1 and 8.2, can be misleading. The mostone can say is that if points are clearly separated in an (E, P, D) plot (for example,in Figure 8.1 E1 equilibrium is outside the recurrent set), they are also separated in


(a) (b)

Fig. 8.2. Two projections of the (E, P, E) representation of the flow. E1 (red), E2 (green),E3 (blue), heteroclinic connections from E2 to E3 (green), from E1 to E3 (red) and from E3 toE2 (shades of blue), superimposed over a generic long-time ‘turbulent’ trajectory (grey). (a) As inFigure 8.2 (b), with labels omitted for clarity. (b) A plot of E = P −D yields a clearer visualizationthan Figure 8.2 (a). System size L = 22.

the full state space. Converse is not true – states of very different topology can havesimilar energies.

An example is the relative periodic orbit (Tp, ℓp) = (32.8, 10.96) (see Figure 6.1 (b))which is the least unstable short relative periodic orbit we have detected in this sys-tem. It appears to be well embedded within the turbulent flow. The mean power Pp

evaluated as in (3.11), see Figure 8.1, is numerically quite close to the long-time tur-bulent time average P . Similarly close prediction of mean dissipation rate in the planeCouette flow from a single-period periodic orbit computed by Kawahara and Kida [31]has lead to optimistic hopes that ‘turbulence’ is different from low-dimensional chaos,insofar that the determination of one special periodic orbit could yield all long-timeaverages. Regrettably, not true – as always, here too one needs a hierarchy of periodicorbits of increasing length to obtain accurate predictions [11].

For any given relative periodic orbit a convenient visualization is offered by themean velocity frame, that is, a reference frame that rotates with velocity cp = ℓp/Tp.In the mean velocity frame a relative periodic orbit becomes a periodic orbit, as inFigure 8.3 (b). However, each relative periodic orbit has its own mean velocity frameand thus sets of relative periodic orbits are difficult to visualize simultaneously.

9. Summary. In this paper we study the Kuramoto-Sivashinsky flow as a stag-ing ground for testing dynamical systems approaches to moderate Reynolds numberturbulence in full-fledged (not a few-modes model), infinite-dimensional state spacePDE settings [25], and present a detailed geometrical portrait of dynamics in theKuramoto-Sivashinsky state space for the L = 22 system size, the smallest systemsize for which this system empirically exhibits ‘sustained turbulence.’

Compared to the earlier work [6, 37, 38, 40], the main advances here are the newinsights in the role that continuous symmetries, discrete symmetries, low-dimensionalunstable manifolds of equilibria, and the connections between equilibria play in orga-nizing the flow. The key new feature of the translationally invariant KS on a periodicdomain are the attendant continuous families of relative equilibria (traveling waves)


(a)

v1v2

v3

(b)

v1v2

v3

2

Fig. 8.3. The relative periodic orbit with (Tp, ℓp) = (33.5, 4.04) from Figure 6.1 (c) whichappears well embedded within the turbulent flow: (a) A stationary state space projection, traced forfour periods Tp. The coordinate axes v1, v2, and v3 are those of Figure 5.6; (b) In the co-movingmean velocity frame.

and relative periodic orbits. We have now understood the preponderance of solutionsof relative type, and lost fear of them: a large number of unstable relative periodicorbits and periodic orbits has been determined here numerically.

Visualization of infinite-dimensional state space flows, especially in presence ofcontinuous symmetries, is not straightforward. At first glance, turbulent dynamicsvisualized in the state space appears hopelessly complex, but under a detailed exami-nation it is much less so than feared: for strongly dissipative flows (KS, Navier-Stokes)it is pieced together from low dimensional local unstable manifolds connected by fasttransient interludes. In this paper we offer two low-dimensional visualizations of suchflows: (1) projections onto 2- or 3-dimensional, PDE representation independent dy-namically invariant frames, and (2) projections onto the physical, symmetry invariantbut time-dependent energy transfer rates.

Relative periodic orbits require a reformulation of the periodic orbit theory [10],as well as a rethinking of the dynamical systems approaches to constructing symbolicdynamics, outstanding problems that we hope to address in near future [45, 44]. Whatwe have learned from the L = 22 system is that many of these relative periodic orbitsappear organized by the unstable manifold of E2, closely following the homoclinicloop formed between E2 and τ1/4E2.

In the spirit of the parallel studies of boundary shear flows [23], the Kuramoto-Sivashinsky L = 22 system size was chosen as the smallest system size for which KSempirically exhibits ‘sustained turbulence.’ This is convenient both for the analysis ofthe state space geometry, and for the numerical reasons, but the price is high - muchof the observed dynamics is specific to this unphysical, externally imposed periodicity.What needs to be understood is the nature of equilibrium and relative periodic orbitsolutions in the L → ∞ limit, and the structure of the L = ∞ periodic orbit theory.

In summary, Kuramoto-Sivashinsky (and plane Couette flow, see ref. [19]) equi-libria, relative equilibria, periodic orbits and relative periodic orbits embody Hopf’svision [26]: together they form the repertoire of recurrent spatio-temporal patternsexplored by turbulent dynamics.


Acknowledgments. We are grateful to Y. Lan for pointing out to us the exis-tence of the E1 equilibrium at the L = 22 system size, J. Crofts for a key observation[8] that led to faster relative periodic orbit searches, J.F. Gibson for many spiritedexchanges, and the anonymous referee for many perspicacious observations. P.C. andE.S. thank G. Robinson, Jr. for support. E.S. was partly supported by NSF grantDMS-0807574. R.L.D. gratefully acknowledges the support from EPSRC under grantGR/S98986/01.

Appendix A. Integrating Kuramoto-Sivashinsky equation numerically.

The Kuramoto-Sivashinsky equation in terms of Fourier modes:

uk = F [u]k =1

L

∫ L

0

u(x, t)e−iqkxdx , u(x, t) = F−1[u] =∑

k∈Z

ukeiqkx (A.1)

is given by

˙uk =(

q2k − q4

k

)

uk − iqk

2F [(F−1[u])2]k . (A.2)

Since u is real, the Fourier modes are related by u−k = u∗k.

The above system is truncated as follows: The Fourier transform F is replacedby its discrete equivalent

ak = FN [u]k =N−1∑

n=0

u(xn)e−iqkxn , u(xn) = F−1N [a]n =

1

N

N−1∑

k=0

akeiqkxn , (A.3)

where xn = nL/N and aN−k = a∗k. Since a0 = 0 due to Galilean invariance and

setting aN/2 = 0 (assuming N is even), the number of independent variables in thetruncated system is N − 2:

ak = vk(a) =(

q2k − q4

k

)

ak − iqk

2FN [(F−1

N [a])2]k , (A.4)

where k = 1, . . . , N/2 − 1, although in the Fourier transform we need to use ak overthe full range of k values from 0 to N − 1. As ak ∈ C, (A.4) represents a system ofordinary differential equations in RN−2.

The discrete Fourier transform FN can be computed by FFT. In Fortran and C,the FFTW library ref. [14] can be used.

In order to find the fundamental matrix of the solution, or compute Lyapunovexponents of the Kuramoto-Sivashinsky flow, one needs to solve the equation for adisplacement vector b in the tangent space:

b =∂v(a)

∂ab . (A.5)

Since FN is a linear operator, it is easy to show that

bk =(

q2k − q4

k

)

bk − iqkFN [F−1N [a] ⊗F−1

N [b]]k , (A.6)

where ⊗ indicates componentwise product of two vectors, that is, a⊗ b = diag(a) b =diag(b) a. This equation needs to be solved simultaneously with (A.4).


Equations (A.4) and (A.6) were solved using the exponential time differencing4th-order Runge-Kutta method (ETDRK4) [7, 30].

Appendix B. Determining stability properties of equilibria, travelingwaves, and relative periodic orbits.

Let f t be the flow map of the Kuramoto-Sivashinsky equation, that is f t(a) = a(t)is the solution of (A.4) with initial condition a(0) = a. The stability properties of thesolution f t(a) are determined by the fundamental matrix J(a, t) consisting of partialderivatives of f t(a) with respect to a. Since a and f t are complex valued vectors,the real valued matrix J(a, t) contains partial derivatives evaluated separately withrespect to the real and imaginary parts of a, that is

J(a, t) =∂f t(a)

∂a=

∂ftR,1

∂aR,1

∂ftR,1

∂aI,1

∂ftR,1

∂aR,2

∂ftI,1

∂aR,1

∂ftI,1

∂aI,1

∂ftI,1

∂aR,2· · ·

∂ftR,2

∂aR,1

∂ftR,2

∂aI,1

∂ftR,2

∂aR,2

.... . .

(B.1)

where ak = aR,k + iaI,k and f tk = f t

R,k + if tI,k. The partial derivatives ∂ft

∂aR,jand ∂ft

∂aI,j

are determined by solving (A.6) with initial conditions bk(0) = bN−k(0) = 1 + 0i andbk(0) = −bN−k(0) = 0 + 1i, respectively, for k = j and bk(0) = 0 otherwise.

The stability of a periodic orbit with period Tp is determined by the location ofeivenvalues of J(ap, Tp) with respect to the unit circle in the complex plane.

Because of the translation invariance, the stability of a relative periodic orbit isdetermined by the eigenvalues of the matrix g(ℓp)J(ap, Tp), where g(ℓ) is the actionof the translation operator introduced in (2.6), which in real valued representationtakes the form of a block diagonal matrix with the 2 × 2 blocks

(

cos qkℓ sin qkℓ− sin qkℓ cos qkℓ

)

, k = 1, 2, . . . , N/2 − 1 .

For an equilibrium solution aq, f t(aq) = aq and so the fundamental matrix J(aq, t)can be expressed in terms of the time independent stability matrix A(aq) as follows

J(aq, t) = eA(aq)t,

where

A(aq) =∂v

∂a

∣

∣

∣

∣

a=aq

. (B.2)

Using the real valued representation of (B.1), the partial derivatives of v(a) withrespect to the real and imaginary parts of a are given by

∂vk

∂aR,j=

(

q2k − q4

k

)

δkj − iqkFN [F−1N [a] ⊗F−1

N [b(j)R ]]k ,

∂vk

∂aI,j=

(

q2k − q4

k

)

iδkj − iqkFN [F−1N [a] ⊗F−1

N [b(j)I ]]k , (B.3)

where b(j)R and b

(j)I are complex valued vectors such that b

(j)R,k = b

(j)R,N−k = 1 + 0i and

b(j)I,k = −b

(j)I,N−k = 0 + 1i for k = j and b

(j)R,k = b

(j)I,k = 0 otherwise. In terms of aR,k


and aI,k we have

∂vR,k

∂aR,j=

(

q2k − q4

k

)

δkj + qk(aI,k+j + aI,k−j) ,

∂vR,k

∂aI,j= −qk(aR,k+j − aR,k−j) , (B.4)

∂vI,k

∂aR,j= −qk(aR,k+j + aR,k−j) ,

∂vI,k

∂aI,j=

(

q2k − q4

k

)

δkj − qk(aI,k+j − aI,k−j) ,

where δkj is Kronecker delta.The stability of equilibria is characterized by the sign of the real part of the

eigenvalues of A(aq). The stability of a relative equilibrium is detemined in the co-moving reference frame, so the fundamental matrix takes the form g(cqt)J(aq, t). Thestability matrix of a relative equilibrium is thus equal to A(aq)+cqL where L = iqkδkj

is the Lie algebra translation generator, which in the real-space representation takesthe form L = diag(0, q1, 0, q2, . . .).

Appendix C. Levenberg–Marquardt searches for relative periodic or-bits.

To find relative periodic orbits of the Kuramoto-Sivashinsky flow, we use multipleshooting and the Levenberg–Marquardt (LM) algorithm implemented in the routinelmder from the MINPACK software package [43].

In order to find periodic orbits, a system of nonlinear algebraic equations needs tobe solved. For flows, this system is underdetermined, so, traditionally, it is augmentedwith a constraint that restricts the search space to be transversal to the flow (oth-erwise, most of the popular solvers of systems of nonlinear algebraic equations, e.g.those based on Newton’s method, cannot be used). When detecting relative periodicorbits, a constraint is added for each continuous symmetry of the flow. For example,when detecting relative periodic orbits in the complex Ginzburg Landau equation,Lopez et al. [40] introduce three additional constraints.

Our approach differs from those used previously in that we do not introduce theconstraints. Being an optimization solver, the LM algorithm has no problem withsolving an underdetermined system of equations, and, even though lmder explicitlyrestricts the number of equations to be not smaller than the number of variables, theadditional equations can be set identically equal to zero [8]. In fact, there is numericalevidence that, when implemented with additional constraints, the solver usually takesmore steps to converge from the same seed, or fails to converge at all [8]. In whatfollows we give a detailed description of the algorithm and the search strategy whichwe have used to find a large number of relative periodic orbits defined in (2.20) andpre-periodic orbits defined in (2.21).

When searching for relative periodic orbits of truncated Kuramoto-Sivashinskyequation (A.4), we need to solve the system of N − 2 equations

g(ℓ)fT(a) − a = 0 , (C.1)

with N unknowns (a, T, ℓ), where f t is the flow map of the Kuramoto-Sivashinskyequation. In the case of pre-periodic orbits, the system has the form

− g(−ℓ)[fT(a)]∗ − a = 0 , (C.2)


(see (7.1)).We have tried two different implementations of the multiple shooting. The em-

phasis was on the simplicity of the implementations, so, even though both implemen-tations worked equally well, each of them had its own minor drawbacks.

In the first implementation, we fix the total number of steps within each shootingstage and change the numerical integrator step size h in order to adjust the totalintegration time to a desired value T.

Let (a, T, ℓ) be the starting guess for a relative periodic orbit obtained througha close return within a chaotic attractor (see below). We require that the initialintegration step size does not exceed h0, so we round off the number of integrationsteps to n = ⌈T/h0⌉, where ⌈x⌉ denotes the nearest integer larger than x.

The integration step size is equal to h = T/n. With the number of shooting stagesequal to m, the system in (C.1) is rewritten as follows

F (1) =f τ (a(1)) − a(2) = 0 ,

F (2) =f τ (a(2)) − a(3) = 0 ,

· · · (C.3)

F (m−1) =f τ (a(m−1)) − a(m) = 0 ,

F (m) =g(ℓ)f τ ′

(a(m)) − a(1) = 0 ,

where τ = ⌊n/m⌋h (⌊x⌋ is the nearest integer smaller than x), τ ′ = nh − (m − 1)τ ,and a(j) = f (j−1)τ (a), j = 1, . . . , m. For the detection of pre-periodic orbits, the lastequation in (C.3) should be replaced with

F (m) = −g(−ℓ)[f τ ′

(a(m))]∗ − a(1) = 0 .

With the fundamental matrix of (C.3) written as

J =

(

∂F (j)

∂a(k)

∂F (j)

∂T

∂F (j)

∂ℓ

)

, j, k = 1, . . . , m , (C.4)

the partial derivatives with respect to a(k) can be calculated using the solution of(A.6) as described in appendix B. The partial derivatives with respect to T are givenby

∂F (j)

∂T=

∂fτ (a(j))∂τ

∂τ∂T = v(f τ (a(j)))⌊n/m⌋/n , j = 1, . . . , m − 1

g(ℓ)v(f τ ′

(a(j)))(1 − m−1n ⌊n/m⌋), j = m .

(C.5)

Note that, even though ∂f t(a)/∂t = v(f t(a)), it should not be evaluated using theequation for the vector field v. The reason is that, since the flow f t is approximatedby a numerical solution, the derivative of the numerical solution with respect to thestep size h may differ from the vector field v, especially for larger step sizes. Weevaluate the derivative by a forward difference using numerical integration with stepsizes h and h + δ:

∂f jh(a)

∂t=

1

jδ

[

f j(h+δ)(a) − f jh(a)]

, j ∈ Z+ (C.6)

with t = jh and δ = 10−7 for double precision calculations. Partial derivatives∂F (j)/∂ℓ are all equal to zero except for j = m, where it is given by

∂F (m)

∂ℓ=

dg

dℓf τ ′

(a(m)) = diag(iqkeiqk ℓ)f τ ′

(a(m)) . (C.7)


Fig. C.1. Numbers of detected relative periodic orbits (RPOs) and pre-periodic orbits (PPOs)with periods smaller than T . The lines indicate the linear fit to the logarithm of the number of orbitsas functions of T in the range T ∈ [70, 120].

This fundamental matrix is supplied to lmder augmented with two rows of zeroscorresponding to the two identical zeros augmenting (C.3) in order to make the numberof equations formally equal to the number of variables, as discussed above.

In the second implementation, we keep h and τ fixed and vary only τ ′ = T− (m−1)τ . In this case, we need to be able to determine the numerical solution of Kuramoto-Sivashinsky equation not only at times tj = jh, j = 1, 2, . . ., but at any intermediatetime as well. We do this by a cubic polynomial interpolation through points f tj (a)and f tj+1(a) with slopes v(f tj (a)) and v(f tj+1(a)). The difference from the firstimplementation is that partial derivatives ∂F (j)/∂T are zero for all j = 1, . . . , m − 1,except for

∂F (m)

∂T= g(ℓ)v(f τ ′

(a(m))) . (C.8)

which, for consistency, needs to be evaluated from the cubic polynomial, not from theflow equation evaluated at f τ ′

(a(m)).For detecting relative periodic orbits of the Kuramoto-Sivashinsky flow with L =

22, we used N = 32, h = 0.25 (or h0 = 0.25 within the first implementation), anda number of shooting stages such that τ ≈ 40.0. While both implementations wereequally successful in detecting periodic orbits of Kuramoto-Sivashinsky flow, we foundthe second implementation more convenient.

The following search strategy was adopted: The search for relative periodic orbitswith T ∈ [10, 200] was conducted within a rectangular region containing the chaoticattractor. To generate a seed, a random point was selected within the region and theflow (A.4) was integrated for a transient time t = 40, sufficient for an orbit to settleon the attractor at some point a. This point was taken to be the seed location. In


order to find orbits with different periods, the time interval [10, 200] was subdividedinto windows of length 10, i.e. [tmin, tmax], where tmin = 10j and tmax = 10(j + 1),

with j = 1, 2, . . . , 19. To determine the seed time T and shift ℓ, we located anapproximate global minimum of ‖g(ℓ)f t(a) − a‖ (or of ‖ − g(−ℓ)[f t(a)]∗ − a‖ in thecase of pre-periodic orbits) as a function of t ∈ [tmin, tmax] and ℓ ∈ (−L/2, L/2]. Wedid this simply by finding the minimum value of the function on a grid of points withresolution h in time and L/50 in ℓ.

Approximately equal numbers of seeds were generated for the detection of relativeperiodic orbits and pre-periodic orbits and within each time window. The hit rate,i.e. the fraction of seeds that converged to relative periodic orbits or pre-periodicorbits, varied from about 70% for windows with tmax ≤ 80 to about 30% for windowswith tmin ≥ 160. The total number of hits for relative periodic orbits and pre-periodicorbits was over 106 each. Each newly found orbit was compared, after factoring outthe translation and reflection symmetries, to those already detected. As the searchprogressed, we found fewer and fewer new orbits, with the numbers first saturatingfor smaller period orbits. At the end of the search we could find very few new orbitswith periods T < 120. Thus we found over 30 000 distinct prime relative periodicorbits with ℓ > 0 and over 30 000 distinct prime pre-periodic orbits with T < 200.

In Figure C.1 we show the numbers of detected relative periodic orbits and pre-periodic orbits with periods less than T . It shows that the numbers of relative periodicorbits and pre-periodic orbits are approx. equal and that they grow exponentially withincreasing T up to T ∼ 130, so that we are mostly missing orbits with T > 130. Thestraight line fits to the logarithm of the numbers of orbits in the interval T ∈ [70, 120],represented by the lines in Figure C.1, indicate that the total numbers of relativeperiodic orbits and pre-periodic orbits with T < 200 could be over 105 each.

To test the structural stability of the detected orbits and their relevance to thefull Kuramoto-Sivashinsky PDE, the numerical accuracy was improved by increasingthe number of Fourier modes (N = 64) and reducing the step size (h = 0.1). Only ahandful of orbits failed this higher-resolution test. These orbits were not included inthe list of the 60,000+ orbits detected.

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